A toric localization similar to (REF ) exists also for K-theoretic DT invariants as defined in [1]}. The K-theoretic DT invariants are a refinement of numerical DT invariants defined for projective moduli spaces with symmetric obstruction theories, which was developed parallel to the motivic and cohomological refinement of Kontsevich-Soibelman and Joyce and collaborators. One expect in general that they correspond to the \(\chi _y\)  genus of the Hodge polynomial of cohomological DT theory, hence in particular one replace \(\mathbb {L}^{1/2}\)  by \(-y\)  in K-theoretic formulas. When the moduli space \(X\)  is non compact, but has a \(\mathbb {C}^\ast \) -action with compact fixed components, it was suggested in [1]} to use the equation (REF ) to define the \(K\)  theoretic invariants of \(X\) . However, this definition depends on the choice of the \(\mathbb {C}^\ast \) -action (this choice is called a choice of slope). The equation (REF ) in the non-projective case explains the origin of this ambiguity: one computes by toric localization only the virtual cohomology of the attracting variety, which is not the whole moduli space and depends on the chosen \(\mathbb {C}^\ast \) -action.
